Exploring the $\beta$ symmetry of supergravity

Kaluza-Klein reductions of low energy string effective actions possess a continuous $O(d,d)$ symmetry. The non-geometric elements of this group, parameterized by a bi-vector $\beta$, are not inherited from the symmetries of the higher-dimensional theory, but constitute instead a symmetry enhancement produced by the isometries of the background. The realization of this enhancement in the parent theory was recently defined as $\beta$ symmetry, a powerful tool that allows to avoid the field reparameterizations of the Kaluza-Klein procedure. In this paper we further explore this symmetry and its impact on the first order $\alpha'$-corrections. We derive the $\beta$ transformation rules from the frame formulation of Double Field Theory (DFT), and connect them to the corresponding rules in the Metsaev-Tseytlin and Bergshoeff-de Roo supergravity schemes. It follows from our results that $\beta$ symmetry is a necessary condition for the uplift of string $\alpha'$-expansions to DFT.